$f(x, y) = x^6 y$ $\dfrac{\partial^2 f}{\partial x^2} = $
Explanation: Taking a second order partial derivative is like taking a regular second order derivative. We take the partial derivative once, then we take another partial derivative. $\dfrac{\partial^2 f}{\partial x^2} = \dfrac{\partial}{\partial x} \left[ \dfrac{\partial f}{\partial x} \right]$ Let's differentiate! $\begin{aligned} \dfrac{\partial^2 f}{\partial x^2} &= \dfrac{\partial}{\partial x} \left[ \dfrac{\partial}{\partial x} \left[ x^6 y \right] \right] \\ \\ &= \dfrac{\partial}{\partial x} \left[ 6x^5 y \right] \\ \\ &= 30x^4 y \end{aligned}$ Therefore, $\dfrac{\partial^2 f}{\partial x^2} = 30x^4 y$.